where \({\underline{\underline{E}}}G\)  is the classifying space of \(G\)  with respect to the family of virtually cyclic groups,
\(H_*^G(-,\mathbf {K} (R))\)  is the \(G\) -homology theory defined in Section 1 of [1]} and \(K_n(RG)\)  stands for the \(n\) -th group of algebraic \(K\) -theory of the group ring \(RG\) .
The Farrell-Jones conjecture predicts that the assembly map is an isomorphism.
The conjecture has been verified for a big family of groups (see [2]} for an excellent survey), and no counterexample has been found so far.
The philosophy in this context is, for a group for which the assembly map is known to be an isomorphism, to perform computations in the topological side in order to extract information about the algebraic \(K\) -theory of the group ring. It is remarkable that the latter are extremely difficult to compute, and at the same time encode fundamental invariants of manifolds, including obstructions to the existence of cobordisms and information about groups of pseudoisotopies. Explicit calculations in this context can be found in [3]}, [4]}, [5]} or [6]}, for example.
